In the structure of the educational course "Algebra" in secondary school, the central place is occupied by the following content-methodological lines: "Numbers and Computations,"; "Algebraic Expressions,"; "Equations and Inequalities,"; "Functions,"; "Logic and Sets." Each of these
content-methodological lines develops over the course of three years of study,
naturally intertwining and interacting with other lines. During the course,
students are required to engage in logical reasoning and use set-theoretic
language. Therefore, the basics of logic are included in the curriculum,
permeating all the main sections of mathematical education and contributing to
students' mastery of the foundations of a universal mathematical language.
Thus, it can be stated that the substantive and structural feature of the
"Algebra" course is its integrated nature.
The same approach is used to
develop the concept of "function": introduction to any new function
in grades 7-8 starts with the consideration of practical problems, the
generalized description of which it represents. From the 9th grade onwards, functions
are introduced based on the internal logic of the development of mathematical
theory. The content of the functional-graphical line is aimed at providing
students with knowledge of functions as the most important mathematical model
for describing and studying various processes and phenomena in nature and
society. A feature of the functional line of the course is the possibility of an in-depth study of the topic "Function." This is facilitated by the
strong propaedeutics of this concept, which begins as early as elementary
school. Already in the 6th grade, students gain an understanding of the concept
of "functional dependency", which allows them to work with the
concept of "function" at a conscious level by the 7th grade. Up to
the 9th grade, students repeatedly return to and refine this concept. By the
end of the 9th grade, students develop an understanding of a function as an
abstract rule of matching elements of two sets of arbitrary nature. Students
begin to familiarize themselves with the properties of functions, initially
considering them for each function they study individually. In the 9th grade,
these properties are generalized for the general concept of a function and used
in graph construction.
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Algebra is one of the foundational courses in secondary school: it provides the basis for the study of other disciplines in both the natural sciences and humanities cycles, and its mastery is necessary for further education and everyday life. Developing students' scientific understanding of the origin and essence of algebraic abstractions, the way mathematical science reflects phenomena and processes in nature and society, and the role of mathematical modeling in scientific knowledge and practice contributes to the formation of a scientific worldview and the quality of thinking necessary for adaptation in a modern digital society. Studying algebra naturally promotes the development of skills such as observation, comparison, and finding patterns and requires critical thinking, the ability to justify actions and conclusions, and formulate statements.
Mastering the algebra course fosters the development of students' logical thinking: they use deductive and inductive reasoning, generalization and specification, abstraction, and analogy. Teaching algebra involves a significant amount of independent work by students, so independent problem-solving naturally embodies the activity principle of learning. This contributes to the formation of intellectual honesty and objectivity, the ability to overcome mental stereotypes arising from everyday experience, and is oriented towards the upbringing of personal qualities that ensure social mobility and the ability to make independent decisions.
The content of the "Numbers and Computations" line serves as the foundation for further study of mathematics, promotes the development of logical thinking in students, the formation of the ability to use algorithms, and the acquisition of practical skills necessary for everyday life. The development of the concept of a number in basic school is associated with rational and irrational numbers, forming ideas about real numbers. Here, students also have the opportunity to study more complex arithmetic topics (Euclidean algorithm, fundamental theorem of arithmetic). The completion of the mastery of the numerical line is assigned to the upper level of general education.
The content of the algebraic lines "Algebraic Expressions" and "Equations and Inequalities" contributes to the formation of students' mathematical tools necessary for solving mathematics problems, related subjects, and practical-oriented tasks in the surrounding reality. In primary school, educational material is grouped around rational expressions, and in this course, examples of irrational expressions are also introduced. Algebra demonstrates the importance of mathematics as a language for constructing mathematical models and describing processes and phenomena of the real world. The objectives of algebra education also include further development of algorithmic thinking, which is necessary for mastering the computer science course and acquiring skills in deductive reasoning. Transforming symbolic forms makes a specific contribution to the development of imagination and abilities for mathematical creativity.
In the course, students have the opportunity to become acquainted with trigonometric expressions and their transformations. A feature of the algebraic line of the BYOM course for grades 7-9, which enhances its practical orientation, is that students are initially offered to solve a practical problem. During the construction or solution of the mathematical model for this problem, they came to the necessity of expanding their existing mathematical tools. This approach is used when introducing all major types of equations (linear Diophantine equations, linear equations with two unknowns and their systems, quadratic equations, fractional-rational equations), as well as systems and sets of inequalities. This allows students to see the connection between the "lifeless" letters of algebra and the "living" world around them and is one way to motivate high school students to study mathematics.
With the same goal, the course discusses a large number of physical problems, the solutions of which are reduced to the methods and techniques just studied, thereby forming students' understanding of mathematics as a powerful tool for understanding real-world processes.
Studying this material contributes to the development of students' ability to use various expressive means of mathematical language — verbal, symbolic, graphical — and also contributes to shaping their understanding of the role of mathematics in the development of civilization and culture.
The educational material of the section "Logic and Sets" aims at the mathematical development of students, forming their ability to express thoughts accurately, concisely, and clearly in oral and written speech.
Thus, the main goal of this course is to develop student's learning skills, their intellectual and moral development and upbringing, the preservation and support of health, and the mastery of each student along an individual trajectory of self-development with a system of deep and solid mathematical knowledge, skills, and abilities necessary for further education in any profile of high school and educational institutions of secondary vocational education.
The square root of a number. Concept of an irrational number. Decimal approximations of irrational numbers. Properties of arithmetic square roots and their application to transforming numerical expressions and calculations. Real numbers. Identity of the form
a2=a(√a)2, where a ≥ 0; a2=a√(a2)=|a|. Exponents with integer exponents and their properties. Standard notation of a number.
Quadratic trinomial; factoring a quadratic trinomial. Algebraic fraction. The basic property of an algebraic fraction. Addition, subtraction, multiplication, and division of algebraic fractions. Rational expressions and their transformation.
Quadratic equation, incomplete quadratic equations. Quadratic equation roots formula. Vieta's theorem. Solving equations reducible to linear and quadratic. Biquadratic equations. Simple fractional-rational equations. Quadratic equations with parameters. Graphical interpretation of equations with two variables and systems of linear equations with two variables. Examples of solving systems of nonlinear equations with two variables. Systems of two linear equations with absolute values.
Solving word problems algebraically and arithmetically. Problems reducible to solving quadratic equations.
Numerical inequalities and their properties. Inequality with one variable. Equivalence of inequalities. Proof of inequalities. Some notable inequalities. Linear inequalities with one variable. Systems of linear inequalities with one variable. Sets of linear inequalities with one variable. Linear inequalities with two variables and their systems. Graphical representation of the set of their solutions. Quadratic inequalities. Solving rational inequalities. Interval method.
Concept of a function. Domain and range of a function. Ways of defining functions. Graph of a function. Reading properties of a function from its graph. Examples of graphs of functions reflecting real processes. Functions describing direct and inverse proportional relationships and their graphs. Functions y=x, y=x2, y=x3, y=√x, y=|x|. Graphical solution of equations and systems of equations.
Piecewise-defined functions. Functions y=ax2, y=ax2+b, y=k(x−d)2, y=ax2, y=ax2+b, y=k(x-d)2 and their graphs. Quadratic function y=ax2+bx+c.
Necessity and sufficiency. Properties and characteristics. Criteria. Complex statements.
Set-theoretical concepts. Set, element of a set. Defining sets by listing elements by characteristic property. Standard notation of numerical sets. Empty set and its notation. Subset. Union and intersection of sets. Illustration of relationships between sets using Euler-Venn diagrams.
Planned subject outcomes for mastering the content of the course
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The student will learn to:
• use initial concepts of the set of real numbers for comparison, rounding, and calculations; • represent real numbers by points on the number line;
• apply the concept of arithmetic square root;
• find square roots, using a calculator if necessary.
• perform transformations of expressions containing square roots using the properties of roots;
• use notation for large and small numbers using decimal fractions and powers of 10.
The student will have the opportunity to learn:
• apply the identity a2=a(√a)2, where a ≥ 0; a2=a(√a)2=|a|, for transformations of expressions with roots;
• calculate the approximate value of the square root using Newton's Method;
• prove properties of arithmetic square roots;
• transform expressions of the form a+bc√(a+b√c) .
The student will learn to:
• apply the concept of exponentiation with integer exponents, perform transformations of expressions containing integer exponents;
• perform algebraic manipulations of rational expressions based on rules of operations with polynomials and algebraic fractions;
• factorize quadratic trinomials;
• apply expression transformations to solve various problems from mathematics, related subjects, and real-life scenarios.
The student will have the opportunity to learn:
• perform polynomial division using the long division method;
• perform transformations of rational expressions, isolating the integer part of the fraction.
The student will learn to:
• solve linear, quadratic equations and rational equations reducible to them, systems of two equations with two variables;
• conduct elementary investigations of equations and systems of equations, including using graphical representations (determine whether an equation or system of equations has solutions, and if so, how many, etc.).
• transition from verbal formulation of a problem to its algebraic model by composing an equation or system of equations, interpreting the obtained result in accordance with the context of the problem;
• apply properties of numerical inequalities for comparison and estimation;
• solve linear inequalities with one variable and their systems;
• provide a graphical illustration of the solution set of an inequality system of inequalities. The student will have the opportunity to learn:
• independently construct and use algorithms for solving the studied cases of word problems;
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• solve non-standard problems using an initial plan;
• solve problems by transitioning from the verbal formulation of the problem condition to the algebraic model by composing systems of linear equations with three or more variables;
• determine the number of solutions of the system analytically;
• investigate systems of equations with two variables containing letter coefficients;
• solve systems of two linear equations with two variables with absolute values;
• apply the method of addition and algebraic addition for systems with three or more variables;
• solve systems of linear and quadratic inequalities, quadratic inequalities;
• solve systems of linear inequalities with one variable with absolute values;
• plot on the coordinate plane sets of points defined by inequalities with two variables and their systems, systems of inequalities with absolute values;
• apply the known method of completing the square of a trinomial to derive the general formula for the roots of a quadratic equation;
• use Vieta's theorem to perform various tasks;
• apply special techniques for computing the roots of a quadratic equation;
• investigate linear and quadratic equations with letter coefficients;
• investigate quadratic inequalities with letter coefficients;
• solve fractional-rational equations by substitution and extracting the integer part;
• solve integer and fractional-rational inequalities by interval method;
• apply the mean inequality to find the maximum (minimum) value of a polynomial;
• prove inequalities using various methods.
The student will learn:
• to understand and use functional concepts and language (terms, symbolic notations);
• to determine the value of a function given the value of the argument;
• to determine the properties of a function from its graph;
• to plot graphs of elementary functions such as y=x2, y=x3, y=x, y=x, y=x2, y=x3, y=√x, y=|x|;
• to describe the properties of a numerical function from its graph.
The student will have the opportunity to learn:
• to transition from one method of defining a function to another;
• to compare the properties of different functions;
• to plot and interpret graphs of piecewise-defined functions;
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• to plot and interpret graphs of functions of the form y=ax2, y=ax2+h, y=k(x−d)2, y=ax2, y=ax2+h, y=k(x-d)2;
• to find the minimum and maximum values of a quadratic trinomial on a given interval;
• to plot and interpret the graph of a quadratic function y=ax2+bx+c.
Logic and Sets
The student will have the opportunity to learn:
• to compose, read, and write complex statements (implications) and their inverses;
• to distinguish between attributes and properties;
• to differentiate between properties, attributes, and criteria;
• to determine and justify the truth or falsity of complex statements;
• to compose, read, and write complex statements using logical connectives "and," "or";
• to find the intersection or union of numerical intervals when solving systems and sets of inequalities;
• to find the intersection and union of sets, the complement and difference of sets;
• to define sets by enumerating their elements by characteristic properties;
• to use standard notations for numerical sets;
• to apply the concepts of equal sets, correspondence between sets, one-to-one correspondence between sets, and equivalent sets;
• to illustrate relationships between sets using Euler-Venn diagrams.
•to construct the conjunction and disjunction of statements and use mathematical symbolism to represent them;
• apply De Morgan's laws;
• prove the countability or uncountability of sets;
• prove properties of operations on sets, including De Morgan's laws formulas.
During lessons introducing new knowledge, during independent study tasks, and when completing tasks at a creative level, only success is evaluated, while errors are identified and corrected based on determining their causes (i.e., rules, algorithms, definitions that are insufficiently understood). During reflection lessons, self-assessment is used, and marks in the journal are assigned at the student's discretion. Marks for tests are given to all students, with the difficulty level chosen so that approximately 75% of the class can achieve grades ‘A’ and ‘B’. Starting from the 8th grade, students prepare for test-based assessment, for which express tests are offered in the course. Based on the results of the test, students can check and assess their progress, for which a sample for self-assessment and a success scale is provided at the end of each test. Organizing this type of control not only contributes to the formation of subject-specific results but also to meta-subject (mastery of the basics of self-monitoring, self-assessment in educational activities) and personal (responsible attitude towards learning, readiness, and ability of students for self-development and self-education based on motivation for learning and cognition, conscious choice, and development of further individual educational trajectories) results.
Homework consists of two parts:
• The compulsory part includes 2–3 tasks feasible for each student, taking approximately 30 minutes of independent work.
• The optional part consists of 1–2 additional tasks. The compulsory part of the homework is selected by the teacher.
Considering the students' age characteristics, it is recommended that they be involved in selecting homework tasks.
For the optional part of the homework, tasks marked with an asterisk (*) can be used.
Self-checking of the compulsory part of homework, error correction, and marking in notebooks can be done at the beginning of the lesson by the students themselves using a prepared sample provided by the teacher through presentations. In this case, during notebook checks, the teacher evaluates only the correctness of self-checking. It is recommended to individually check the additional part of the homework. Students who correctly solve creative tasks, as assigned by the teacher, submit their solutions on sheets, which are then displayed in the classroom with the surnames of those who correctly solved the proposed tasks. Only positive marks are given when evaluating these tasks.
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Express tests at the end of each section can also be used as homework (the * part serves as an optional part of the homework).
Teaching mathematical language as a specific means of communication in comparison with natural language is one of the most important features of the BYOM program. Proficient mathematical language is evidence of clear and organized thinking. Therefore, mastery of this language, understanding the precise meaning of statements, and the logical connections between them extend to proficiency in natural language, which significantly contributes to the formation and development of students' thinking as a whole. By the beginning of the 8th grade, as part of the logical line, students have become acquainted with concepts such as statements, their negations, types of statements, implications, and the equivalence of statements. They are familiar with the concepts of definition and theorem and know some methods of proving statements (including proof by contradiction). They can use quantifiers, implication signs, and equivalence signs.
In the 8th grade, the work on mastering the mathematical language by students continues. In the first chapter, they gain an understanding of the following types of statements: property, attribute, and criterion. They learn about the mathematical meaning of the concepts of "necessity" and "sufficiency" and their use in implications. This educational material is of great importance in the study of the related subject of geometry. This interdisciplinary connection is emphasized by the content on which these logical concepts are introduced. Eighth graders expand their understanding of complex statements by familiarizing themselves with statements constructed using logical connectives "and" and "or". Students have the opportunity to become acquainted with such concepts of mathematical logic as conjunction and disjunction, as well as with basic logic formulas. The assimilation of norms for constructing complex statements plays a significant role in organizing children's thinking. The skills developed in the study of this topic (articulate speech and the ability to reason logically) are necessary for students not only in algebra and geometry lessons but also in other subjects and in life.
It should be understood that the
main goal of lessons at the beginning of the academic year is to review
previously studied material. In the first chapter of the 8th grade, new
material is studied using the content of various topics from the mathematics course
of grades 5-6 and algebra 7, which allows for their review in the traditional
format of the BYOM course: parallel to the study of the topic "Language
and Logic." Thus, students have the opportunity to recall previously
studied material, identify and fill possible gaps in knowledge, but at the same
time, "not stand still," but move forward, expanding their
understanding of complex statements. By repeating, something must be added.
Otherwise, learning leads to "intellectual laziness, apathy, and therefore
impedes development." Therefore, when studying the mathematical language,
students review the main topics of the 7th-grade course. They convert common
fractions and mixed numbers into repeating decimals and vice versa. They recall methods for solving linear equations and linear inequalities, reinforce the ability to apply formulas for reduced multiplication to transform expressions, rationalize calculations and factorize, and review various methods of polynomial factorization. They solve word problems. Time is devoted to the concept of a function and its domain, and students practice plotting the graph of a linear function, direct proportion, and piecewise linear function. Students have the
opportunity to review the algorithm for solving linear equations in integers
and methods for solving equations and inequalities involving absolute values.
During the study of this chapter, special attention should be paid to the activation of the algorithm for plotting the graph of a linear function. In this way, students will be prepared for the study of the topics in the next chapter: "Linear equation with two variables and its graph"; "Systems of two linear equations with two variables"; "Graphical solution of systems". To prepare for the study of the topics in the second chapter: "Systems of two linear equations with absolute values" and "Systems of linear inequalities with two variables with absolute values", during the study of the first chapter, attention should be paid to the concept of absolute value and sufficient time should be devoted to solving inequalities and equations with absolute values with one variable.
During the study of the content of the first chapter, students:
• review and systematize previously acquired knowledge;
• apply the learned methods of action to solve problems in typical and exploratory situations;
• justify the correctness of the performed action by referring to the general rule, theorem, property, or definition;
• identify true and false statements, determine and justify their truthfulness and falsity;
• compose, read, and write complex statements;
• construct conjunction and disjunction of statements and use mathematical symbols for their notation.
